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If x^1+x^(−1)=5, what is the value of x^4+x^(−4)?

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If x^1+x^(−1)=5, what is the value of x^4+x^(−4)? [#permalink]

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New post 27 Mar 2017, 03:54
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A
B
C
D
E

Difficulty:

  55% (hard)

Question Stats:

58% (01:28) correct 42% (01:45) wrong based on 121 sessions

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Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)? [#permalink]

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New post 27 Mar 2017, 04:26
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Bunuel wrote:
If x^1 + x^(−1) = 5, what is the value of x^4 + x^(−4)?

A. 527
B. 546
C. 579
D. 600
E. 625


\(x^1 + x^{(−1)} = 5\) ----------------- I


\((x+1/x)^2 = x^2 + 1/x^2 + 2\)

\(5^2 = x^2 + 1/x^2 + 2\) (by using I)

\(x^2 + 1/x^2 = 23\) --------------- II



\((x^2+1/x^2)^2 = x^4 + 1/x^4 + 2\)

\(23^2 = x^4 + 1/x^4 + 2\) (by using II)

\(x^4 + 1/x^4 = 529 - 2 = 527\)

Hence option A is correct
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Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)? [#permalink]

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New post 27 Mar 2017, 11:16
Bunuel wrote:
If x^1 + x^(−1) = 5, what is the value of x^4 + x^(−4)?

A. 527
B. 546
C. 579
D. 600
E. 625


As explained in the above post, the answer will be A.

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Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)? [#permalink]

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New post 29 Mar 2017, 10:10
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Bunuel wrote:
If x^1 + x^(−1) = 5, what is the value of x^4 + x^(−4)?

A. 527
B. 546
C. 579
D. 600
E. 625


We are given that x^1 + x^(−1) = 5, i.e., x + 1/x = 5. We need to determine the value of x^4 + x^(-4), i.e., x^4 + 1/x^4.

Let’s square both sides of the equation x + 1/x = 5.:

(x + 1/x)^2 = 5^2

x^2 + 2(x)(1/x) + 1/x^2 = 25

x^2 + 2 + 1/x^2 = 25

x^2 + 1/x^2 = 23

Now let’s square the above equation:

(x^2 + 1/x^2)^2 = 23^2

x^4 + 2(x^2)(1/x^2) + 1/x^4 = 529

x^4 + 2 + 1/x^4 = 529

x^4 + 1/x^4 = 527

Answer: A
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Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)? [#permalink]

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New post 13 Sep 2017, 04:05
Official Answer

Direct attempts to solve for x in this problem will run into quadratics that don't factor and horrible non-integers that need to be raised to fourth powers. Instead, let's focus on manipulating the equation to solve for x^4 + x^−4 directly.

Given the similar structure of the given information, it seems reasonable to begin by squaring the equation x^1+x^−1=5. Be careful, though, not to simply square each term; exponents do not distribute over addition. Instead, recognize the special quadratic. We're looking at two terms added and then squared, so this expression fits the form (a+b)^2=a^2+2ab+b^2. Thus our result will be


(x^1+x^−1^2=5^2

(x^1)^2+2(x^1)(x^−1)+(x^−1)^2=25


x^2+2+x^−2=25


x^2+x^−2=23


Now just repeat the process of squaring both sides once more:


(x^2+x^−2)^2=23^2


x^4+2(x^2)(x^−2)+x^−4=23^2


x^4+2+x^−4=23^2


x^4+x^−4=23^2−2


And it's not even really necessary to calculate 23^2 (which turns out to be 529). 23^2 must end in a 9, so 23^2−2 must end in a 7, and the answer has to be A.
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Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)?   [#permalink] 13 Sep 2017, 04:05
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